Brian Hamilton

Hexagonal vs. rectilinear grids for explicit finite difference schemes for the two-dimensional wave equation

Finite difference schemes for the 2-D wave equation operating on hexagonal grids and the accompanying numerical dispersion properties have received little attention in comparison to schemes operating on rectilinear grids. This paper considers the hexagonal tiling of the wavenumber plane in order to show that the hexagonal grid is a more natural choice to emulate the isotropy of the Laplacian operator and the wave equation. Performance of the 7-point scheme on a hexagonal grid is better than previously reported as long as the correct stability limit and tiling of the wavenumber plane are taken into account. Numerical dispersion is analysed as a function of temporal frequency to demonstrate directional cutoff frequencies. A comparison to 9-point compact explicit schemes on rectilinear grids is presented using metrics relevant to acoustical simulation. It is shown that the 7-point hexagonal scheme has better computational efficiency than parameterised 9-point compact explicit rectilinear schemes. A novel multiply-free 7-point hexagonal scheme is introduced and the 4-point scheme on a honeycomb grid is discussed.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation as a function of simulation distance

Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%.

Optimised 25-point finite difference schemes for the three-dimensional wave equation

Wave-based methods are increasingly viewed as necessary alternatives to geometric methods for room acoustics simulations, as they naturally capture wave phenomena like diffraction and interference. For methods that simulate the three-dimensional wave equation—and thus solve for the entire acoustic field in an enclosed space—computational costs can be high, so efficient algorithms are critical. In terms of computational complexity, finite difference schemes are possibly the simplest such algorithms, but they are known to suffer from numerical dispersion. High-order and optimised schemes can offer improved numerical dispersion, and thus, computationally efficient numerical solutions. In this paper, we consider two families of explicit finite difference schemes for the second-order wave equation in three spatial dimensions, using 25-point stencils on the Cartesian grid. We review known special cases that lead to high-order accuracy in space (and possibly in time), and we present alternative schemes with opti...

Finite difference schemes in musical acoustics: A tutorial

The functioning of musical instruments is well described by systems of partial differential equations. Whether ones interest is in pure musical acoustics or physical modeling of sound synthesis, numerical simulation is a necessary tool, and may be carried out by a variety of means. One approach is to make use of so-called finite-difference or finite-difference time-domain methods, whereby the numerical solution is computed as a recursion operating over a grid. This chapter is intended as a basic tutorial on the design and implementation of such methods, for a variety of simple systems. The 1-D wave equation and simple difference schemes are covered in Sect. 19.1, accompanied by an analysis of numerical dispersion and stability, as well as implementation details via vector-matrix representations. Similar treatments follow for the case of the ideal stiff bar in Sect. 19.2, the acoustic tube in Sect. 19.3, the 2-D and 3-D wave equations in Sect. 19.4, and finally the stiff plate in Sect. 19.5. Some more general nontechnical comments on more complex extensions to nonlinear systems appear in Sect. 19.6.

Passive time-domain numerical designs for room acoustics simulation

The design of stable time domain numerical simulation methods for room acoustics simulation is a challenging problem. One chief difficulty is in the determination of appropriate stable boundary terminations, particularly when the room geometry is irregular, and when the wall condition is spatially-varying and/or frequency-dependent in a non-trivial way. In this paper, design strategies for stable simulation are presented, based on the finite volume time domain method (FVTD), which, due to its unstructured character, allows for flexible modelling of irregular room geometries. Furthermore, FVTD reduces to the popular finite difference time domain (FDTD) method under certain choices of regular structured mesh. Under locally-reactive wall conditions, the boundary condition can be characterised by a positive real admittance function, variable over the extent of the room boundary. Using frequency-domain analysis techniques, it can be shown that solutions to the complete system are non-increasing. Furthermore, s...